The spin-1 Ising Blume-Emery-Griffiths (BEG) model has been simulated using a cellular automaton (CA) algorithm improved from the Creutz cellular automaton (CCA) for a face-centered-cubic (fcc) lattice. The ground state diagram (k, d) of the fcc BEG model has ferromagnetic (F), quadrupolar (Q), and staggered quadrupolar (SQ) structure regions. The simulations have been made in the SQ region for the parameter values in the intervals -24 <= d = D/J < 0 and -3 <= k = K/J <= 0. The phase diagrams on the (kT(C)/J, d) and the (kT(C)/J, k) planes have been obtained through k = -3 and d = -4 lines, respectively. The SQ region separates into five regions (A(3)B(a), A(3)B(f), AB (type-I), AB (type-II), and AB(3)(f)) which have the different stoichiometric Cu-Au type structures.